We present a Melnikov type approach for establishing transversal intersections of stable/unstable manifolds of perturbed normally hyperbolic invariant manifolds. In our approach, we do not need to know the explicit formulas for the homoclinic orbits prior to the perturbation. We also do not need to compute any integrals along such homoclinics. All needed bounds are established using rigorous computer assisted numerics. Lastly, and most importantly, the method establishes intersections for an explicit range of parameters, and not only for perturbations that are ‘small enough,’ as is the case in the classical Melnikov approach.

Work of Avila and Forni established weak mixing for the generic straight line flow on generic translation surfaces, and the work of Avila and Delecroix determined when weak mixing occurs for the straight line flow on a Veech surface. Following the work of Eskin, Mirzakhani, Mohammadi, which proved that the orbit closure of every translation surface has a very nice structure, one can ask how the orbit closure affects the weak mixing of the straight line flow. In this talk all of the necessary background on translation surfaces and weak mixing will be presented followed by the answer to this question. This is a joint work in progress with Artur Avila and Vincent Delecroix.

Sep 30: No seminar

Oct 7: No seminar

Oct 14: No seminar

A classical property of pseudo-Anosov mapping classes is that
they act on the space of projective measured laminations with
north-south dynamics. This means that under iteration of such a mapping
class, laminations converge exponentially quickly towards its stable
lamination. We will discuss a new construction (joint with Saul
Schleimer) of pseudo-Anosovs where this exponential convergence has base
arbitrarily close to one and so is arbitrarily slow.

For dominant rational maps of compact, complex, Kahler manifolds there is a
conjecture specifying the expected ergodic properties of the map depending on the
relationship between the rates of growth for certain degrees under iteration of the
map. In the present talk, we will discuss the case of two-dimensional Blaschke products,
observing that they fit naturally within this conjecture, having examples from each of the three cases
that the conjecture gives for maps of a surface.
The results to be discussed are included in different works with Mike Shub and Roland Roeder.

Farb and Leininger asked: How many simple closed curves on a finite-type surface $S$ may pairwise intersect at most $k$ times? Przytycki has shown that this number grows at most as a polynomial in $|\chi(S)|$ of degree $k^{2}+k+1$. We present narrowed bounds by showing that the above quantity grows slower than $|\chi(S)|^{3k}$. In particular, the size of a maximal 1-system grows sub-cubically in $|\chi(S)|$. The proof uses a bound for the maximum size of a collection of curves of length at most $L$ on a hyperbolic surface homeomorphic to $S$. Specializing to the case that $S$ is an $n$-holed sphere and $k=2$, we use the coloring computations of Gaster-Greene-Vlamis to show that this bound can be improved to $O(n^5 \log n)$. This is joint work with Tarik Aougab and Ian Biringer.

Periodic orbits play an important role in the study of dynamical systems. In resemblance to the classical Prime Number Theorem in number theory and its relation to the Riemann Hypothesis, it is a natural problem to investigate precise asymptotes for the number of (primitive) periodic orbits as well as the error terms. Such results, known as Prime Orbit Theorems, have been established in many dynamical systems thanks to the works of W. Parry, M. Pollicott, V. Baladi, D. Dolgopyat, C. Liverani, L. Stoyanov, G. A. Margulis, A. Avila, S. Gouëzel, J. C. Yoccoz, M. Tsujii, and many others.

In this talk, we introduce a brief history of such results, focusing mainly on the works of F. Naud, H. Oh, and D. Winter on hyperbolic rational maps. We discuss the main ideas used to obtain such results. If time permits, we discuss how to extend such results to a class of non-hyperbolic rational maps known as (rational) expanding Thurston maps. This is a work-in-progress joint with T. Zheng.

Nov 25: No seminar

Dec 2: Anja Randecker (University of Toronto)
A Class of Infinite Translation Surfaces Where Almost Every Direction is Uniquely Ergodic

Translation surfaces can be obtained from gluing finitely many polygons along parallel edges of the same length. In recent years, people have asked what happens when you glue infinitely instead of finitely many polygons. From that question the field of infinite translation surfaces has evolved. It turns out that the behavior of infinite translation surfaces is in many regards very different and more diverse than the finite case. For instance, Kerckhoff, Masur, and Smillie showed in 1986 that on a finite translation surface the geodesic flow is uniquely ergodic in almost every direction. This is not at all true for infinite translation surfaces in general. However, in this talk, I will introduce a class of infinite translation surfaces where the statement remains true. I will recall the original proof from Kerckhoff, Masur, and Smillie and show how the proof has to be adapted and why that class of infinite translation surfaces was chosen. The presented work is joint with Kasra Rafi.

Consider a $\pi_1$-injective immersion $f:\Sigma \to M$ from a
compact surface $\Sigma$ to a hyperbolic $3$-manifold $(M,h)$. Let $\Gamma$ denote the copy of $\pi_{1}(\Sigma)$ in $\mathrm{Isom}(\mathbb{H}^{3})$ induced by the immersion. In this talk, I will discuss relations between two dynamical quantities: the critical exponent $\delta_{\Gamma}$ and the topological entropy $h_{top}(\Sigma)$ of the geodesic flow for the immersed surface $(\Sigma,f^{*}h)$.
More precisely, when $\Gamma$ is convex cocompact and $\Sigma$ is
negatively curved, there exist two geometric constants $C_{1}(\Sigma,M)$,
$C_{2}(\Sigma,M)\leq 1$ such that $C_{1}(\Sigma,M)\cdot\delta_{\Gamma}\leq h_{top}(\Sigma)\leq C_{2}(\Sigma,M)\cdot\delta_{\Gamma}$.
When $f$ is an embedding, $C_{1}(\Sigma,M)$ and $C_{2}(\Sigma,M)$
are exactly the geodesic stretches (aka Thurston's intersection
numbers) with respect to certain Gibbs measures. Moreover, there are
rigidity phenomena arising from these inequalities. Lastly, if time
permits, I will also discuss applications of these inequalities to
immersed minimal surfaces in hyperbolic $3$-manifolds and derive results
similar to A. Sanders' work on the moduli space of $\Sigma$ introduced
by C. Taubes.

It is a classical result that measurable rigidity holds for $C^2$ conservative Anosov diffeomorphisms. In this talk, I will prove that measurable rigidity is also true for $C^1$ generic conservative Anosov diffeomorphisms. The main ingredient in the proof is the Central Limit Theorem.